Generally they never get paused.  However, contrary to what might seem intuitive, it really doesn't take hours to play the game online against other people.  Most humans v. humans games take less than 2 hours and often less than 1.5 hours.  How is that possible?  When you play single player against 9 AI opponents in free-for-all you have to kill each AI one-by-one on your own.  However, in a team game such as a 4v4 or 5v5 you don't have to kill all of your opponents; you really just match up against one of them and you have 3 or 4 allies to kill the other opponents.  Also, you don't have to go through the boring motions of mopping up defeated opponents.  Normally the players on the losing team will call it "gg" (good games) once it becomes obvious that they are going to lose and the game ends as they quit or surrender.  The other thing that saves time is that most games are played with locked teams on single star maps and with all fast or faster speeds and quick start.  So, it really is possible to finish a 4v4 or 5v5 in under 2 hours.  I think most 1v1's take less than 1:15 to play.

It is possible to save a game and pick it back up later if everyone comes back, however it doesn't happen often in humans v. humans games.  It's mostly real life friends who play comp stomps who do that.  They tend to play 2v6 or 3v7 on huge multistar maps.